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Published for SISSA by Springer Received: June 11, 2020 Accepted: October 26, 2020 Published: November 27, 2020
The large-N limit of the 4d N = 1 superconformal
index
Alejandro Cabo-Bizet,a Davide Cassani,b Dario Martellic,d and Sameer Murthya aDepartment of Mathematics, King’s College London,
The Strand, London WC2R 2LS, U.K. bINFN, Sezione di Padova,
Via Marzolo 8, 35131 Padova, Italy
cDipartimento di Matematica “Giuseppe Peano”, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
dINFN, Sezione di Torino & Arnold-Regge Center, Via Pietro Giuria 1, 10125 Torino, Italy
E-mail: alejandro.cabo_bizet@kcl.ac.uk,davide.cassani@pd.infn.it,
dario.martelli@unito.it,sameer.murthy@kcl.ac.uk
Abstract: We systematically analyze the large-N limit of the superconformal index of N = 1 superconformal theories having a quiver description. The index of these theories is known in terms of unitary matrix integrals, which we calculate using the recently-developed technique of elliptic extension. This technique allows us to easily evaluate the integral as a sum over saddle points of an effective action in the limit where the rank of the gauge group is infinite. For a generic quiver theory under consideration, we find a special family of saddles whose effective action takes a universal form controlled by the anomaly coefficients of the theory. This family includes the known supersymmetric black hole solution in the holographically dual AdS5theories. We then analyze the index refined by turning on flavor chemical potentials. We show that, for a certain range of chemical potentials, the effective action again takes a universal cubic form that is controlled by the anomaly coefficients of the theory. Finally, we present a large class of solutions to the saddle-point equations which are labelled by group homomorphisms of finite abelian groups of order N into the torus.
Keywords: Supersymmetric Gauge Theory, AdS-CFT Correspondence, Conformal Field Theory, Black Holes in String Theory
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Contents
1 Introduction and summary 1
2 Large-N saddles of quiver theories 6
2.1 The elliptic form of the action for N = 1 quiver theories 7
2.2 The saddle-point equations and (m, n) solutions in the continuum limit 8
2.3 The discrete case 11
2.4 The contour deformation 12
3 The effective action of the (m, n) saddle 15
3.1 Evaluation of the action 16
3.2 The τ -independent part of the action 18
3.3 Special families of saddles 19
3.4 Phase structure in the grand-canonical ensemble 23
4 Large-N limit of the refined index 24
4.1 Including flavor chemical potentials 24
4.2 Special families of saddles in the flavored setup 28
5 The effective action written in terms of Tr QIQJQK 29
5.1 A democratic basis of charges 30
5.2 Action controlled by anomalies 31
5.3 Toric quiver gauge theories 34
5.4 Recovering the unflavored action 38
6 More general solutions to the saddle-point equations 40
6.1 Group-theoretic description of the general saddles 40
6.2 Saddles with two factors 43
6.3 N2 string-like saddles 44
6.4 Surface-like saddles 45
7 Outlook 46
A Some useful special functions 47
B Anomaly coefficients and ABJ anomaly cancellation 51
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1 Introduction and summary
The last couple of years have seen good progress in the study of the 161-BPS superconformal index of four dimensional N = 4 super Yang-Mills theory (SYM) and, more generally, the
1
4-BPS index in N = 1 superconformal field theories (SCFT). The index in question is a supersymmetric partition function which receives contributions from states that preserve two supercharges, which is the minimum amount of supersymmetry required to construct such a quantity protected under supersymmetric deformations of the theory. Apart from its importance in capturing the protected spectrum of the field theory, this index also plays an important role in the gauge/gravity duality. The holographic dual of a 4d N = 1 SCFT is a gravitational theory on AdS5 which admits black hole solutions preserving two supercharges [1–5]. The AdS/CFT correspondence predicts that the growth of states of the index in the large central charge limit should capture the Bekenstein-Hawking entropy of the black hole, and it is this aspect that has particularly motivated the recent progress. These indices were first calculated in [6,7] in the form of integrals over unitary matrices, and the recent progress involves a detailed study of these integrals. Independent studies in the last couple of years have reached the conclusion that the 14-BPS index in N = 1 theories (or its direct lifts like the 161-BPS in N = 4 SYM) indeed captures the entropy of the dual black hole at large N [8–22]. The basic idea of all the approaches is the same and can be paraphrased as follows: one calculates the index of the BPS states, and shows that it agrees with the “entropy function” of the BPS black hole. The entropy function is a function of the chemical potentials dual to the charges whose Legendre transform yields the black hole entropy [8]. More precisely, on the gravity side this function is a regularized on-shell action of the dual AdS5 black hole geometry [9,23].
The different studies are essentially variants of three approaches, each of which have advantages and disadvantages. One approach is to study the index in a Cardy-like limit [10,
13, 14,16–18, 22]. In this approach the rank N of the gauge group can be finite, but the disadvantage is that the method only applies in the infinite charge limit, or equivalently, to infinitely large black holes. The advantage is that it applies to generic superconformal theories, and the answer only depends on universal quantities like the conformal anomaly coefficients. Another advantage is that we can apply it to the index involving two inde-pendent angular momenta (presented in (1.2) below). The other two approaches, which we presently discuss, calculate the index involving only one combination of the angular momenta (presented in (1.7) below), although this is a technical limitation which may be possible to overcome.
A second approach is the Bethe-ansatz-like formalism which does not directly use the matrix integral formulation of [6, 7], but instead rewrites the index as a different contour integral which can be performed by a residue calculation. This approach, originally designed for the 3d topologically twisted index [24,25] and the dual AdS4 black holes [26,
27], was developed for 4d, N = 1 theories in [28, 29], and applied to the problem of black hole microstate counting in [12] (for N = 4 SYM) and [19, 20] (for more general toric quiver gauge theories). An advantage of this approach is its regime of applicability, which is that the rank N of the gauge group can be large, while the charge of the states
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can be finite in units of N2, which is exactly the regime of parameters of the black hole solution in supergravity. A practical limitation is that it relies on finding solutions to the associated Bethe-ansatz-like equations, which have not been systematically studied so far. It should be said that some families of solutions for specific theories have been found in [12,22,30,31] and, importantly, this includes a solution corresponding to a black hole. Although in this approach the large-N index takes the form of a sum over solutions to the Bethe-ansatz equations, an interpretation of the latter as saddle-points of the integral is not clear (see [12]).
The third approach, which we use here, is a direct saddle-point analysis of the matrix integral that was developed for N = 4 SYM in [21]. The integral over unitary matrices in [6, 7] reduces, in a completely standard manner, to an integral over the corresponding eigenvalues which live on a circle. The essence of the approach of [21] is to extend the range of eigenvalues of the unitary matrix from a circle to a torus, one of whose cycles is the original circle. This prompts us to refer to this approach as that of elliptic extension. As we review below, this approach allows us to find solutions of the saddle-point equations and, further, it allows us to calculate the effective action at each saddle point in a straightforward manner. In this paper we use this idea to lay down a simple and systematic approach to the calculation of the large-N index of N = 1 quiver theories. We study the basic index, which may be defined for any N = 1 supersymmetric field theory with an R-symmetry, as well as the index refined by including chemical potentials for flavor (non-R) symmetries, and our focus will be to extract simple universal results for generic theories.
In the rest of this introductory section, we present the context of the problem and our main results. We consider N = 1 superconformal theories on S1× S3. The relevant conserved charges are the angular momenta J1, J2 i.e. the Cartan elements of the SO(4) isometry of S3, the energy E generating translations around S1, and the U(1) R-charge Q. There is a choice of supercharge Q that commutes with the bosonic charges J1,2+Q2, and for which
{Q, Q} = E − J1− J2− 3
2Q . (1.1)
The superconformal index, defined as the following trace, I(σ, τ ; n0) = TrH(−1)Fe−β{Q,Q}e2πi(σ−n0)(J1+
Q
2)+2πi τ (J2+
Q
2) (1.2)
is independent of β, as it only gets contributions from the cohomology of Q, namely states that obey the BPS condition E − J1− J2−3
2Q = 0. (For this reason the factor e
−β{Q,Q}is sometimes suppressed.) The chemical potentials σ, τ are allowed to take complex values, with Im(σ), Im(τ ) > 0. The integer parameter n0 was introduced in [9,17] so as to facilitate the comparison with the gravitational results. In particular, in the Cardy-like limit σ, τ → 0 studied in [16, 17], it is n0 = ±1, rather than n0 = 0, that gives the O(N2) black hole entropy. Since (1.2) is only a function of the two variables (σ − n0) and τ , we can reabsorb n0 by a shift of σ, that is
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as long as σ and τ are independent variables. In this paper we study the slice σ = τ in the space of variables, that is we study the index
I(τ, τ ; n0) ≡ I(τ − n0, τ ; 0) . (1.4) After this identification is made, the independent variables are τ and the discrete choice of n0. In fact only n0 = 0 and n0 = 1 give inequivalent choices. This is seen by making the change of variable τ = τ0+n0
2 , so that I τ0+ n0 2 , τ 0+ n0 2 ; n0 = TrH(−1)Fe−β{Q,Q}e2πi τ 0(2J ++Q)−2πi n0J−, (1.5)
where J±= 12(J1±J2) are the Cartan generators of the two SU(2) factors in SO(4). Since J− takes half-integer values, the choice of n0 is only relevant modulo 2. For n0 = 0, we have the usual expression for the index with the two chemical potentials σ and τ identified, that is I(τ, τ ; 0). For n0 = 1, we obtain from (1.2)
I(τ, τ ; 1) ≡ I(τ − 1, τ ; 0) = TrHe−β{Q,Q}e2πi τ (2J++Q)e−πi Q, (1.6)
where we used that e−2πi J1 = (−1)F. Written in this way, the index has the form of a
thermal partition function where τ is a chemical potential for the charge 2J++ Q and (−1)F is replaced by an insertion of e−πi Q, which can be seen as a shift in the R-symmetry chemical potential. This interpretation matches the dual black hole asymptotics, where the supercharge naturally is anti-periodic while transported around the Euclidean time circle [9].1 However, in our discussion we will find it convenient to keep n0 generic, and we will denote the index under study by
I(τ ) = I(τ, τ ; n0) = TrH(−1)Fe−β{Q,Q}e−2πin0(J1+
Q
2)+2πi τ (2J++Q). (1.7)
As mentioned above, the trace (1.2) can be calculated in terms of an integral over unitary matrices. Writing the eigenvalues of a unitary matrix as e2πiui, this can be expressed
as an integral over the gauge variables ui running over the interval [0, 1], this will be the
starting point of our analysis. In this paper we consider N = 1 quiver theories with SU(N ) gauge group at each node of the quiver. The integral then runs over the gauge holonomies of all the gauge groups, this is presented in equation (2.1). The main idea of [21] is to deform the integrand of this integral, without changing its value on the real line, to a complex-valued function defined on the complex u-plane that is periodic under translations by the lattice Zτ + Z. In other words, the integrand is now well-defined on the torus C/(Zτ + Z). In the large-N approximation, we expect that the matrix integral can be written as a sum over solutions to the saddle point equations,
I(τ ) ∼ X
γ∈{saddles}
exp −Seff(τ ; γ)
, (1.8)
1The discussion above also makes it clear that the choices n
0 = −1 and n0 = +1 are related as
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where the saddles γ contributing to the sum are those captured by a certain contour that is a deformation of the original contour along the real axis. This leads to the following questions: what is the complete set of saddles? What is the effective action Seff evaluated on a generic saddle? What is the final contour and what saddles does it pick? As explained in [21], it is a straightforward consequence of the double periodicity mentioned above that the uniform distribution of eigenvalues wrapping the torus C/(Zτ + Z) along any cycle is a saddle-point configuration of the extended integral. Thus we obtain the infinite family of saddles labelled by integers (m, n) corresponding to the cycle wrapped by the string of eigenvalues. The extended integrand itself is governed by a certain special function — the Bloch-Wigner elliptic dilogarithm — which makes the calculation of the action of the saddle points also quite simple. Despite the fact that the extended integrand is not a meromorphic function, for each (m, n) saddle one can show that the original contour can be deformed so as to pass through it, at leading order in the large-N expansion. These remarks answer the first two of the three questions raised above, and we leave the third question for future work. Relatedly, note that although the equation (1.8) gives the complete perturbative expansion around each saddle γ at large N , we do not yet have the exact non-perturbative answer — which would involve making sense of the infinite sum for every value of τ .
With this background and context, we can now describe the main results of this paper. 1. Large-N value of the index
The leading large-N effective action of the (m, n) saddles described above has a simple expression in terms of the third Bernoulli polynomial
Seff(m, n) = πiN2 3m(mτ + n)2 X α∈{multiplets} B3 zα+ (mτ + n)(rα− 1), (1.9)
up to a purely imaginary, τ -independent term that we will discuss later. Here the sum over α runs over all the N = 1 multiplets of the theory with R-charge rα, and
zα= −(n0m + 2n) rα 2 , (1.10)
with {x} = x − bxc being the fractional part of the real number x. The precise expression is given in equation (3.8).
2. Universal gravitational phases
Among all the (m, n) saddles, the saddles having n0m + 2n = 0 or ±1 are special in that their effective action is completely controlled by the R-symmetry anomaly coefficients of the SCFT. (See equations (3.14), (3.18) for the full expressions.) In particular, for n0= ±1 the action of the saddle (m, n) = (1, 0) corresponds precisely to the regularized on-shell action of the supersymmetric black hole in AdS5 [9]. The other solutions in this family also have action proportional to N2. Since, in ad-dition, they only depend on the R-anomaly coefficients, we expect that they should have a universal description as gravitational solutions of the five-dimensional gauged
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supergravity. Independent of its gravitational interpretation, we can use the expres-sion (1.8) to calculate the phase diagram of the theory at large N within this class of solutions. At any given value of τ , the dominant phase is the solution (m, n) which minimizes the real part of the action. We discuss this in section 3.4.
3. Universal form of the action for flavored index
We then study the index refined by adding chemical potentials dual to arbitrary abelian flavor symmetries. This refined index is defined in equation (4.2). The (m, n) configurations are also saddle-points for this index. The effective action as a function of the chemical potentials is cubic. We find that, remarkably, for a particular set of chambers in the space of chemical potentials, the effective action is, once again, com-pletely controlled by the R-symmetry and flavor symmetry anomalies of the theory. In these chambers, the main term in the action is of order N2 and is controlled by the anomaly coefficient CIJ K = Tr(QIQJQK), where QI are certain combinations of
the flavor and R-symmetry charges defined in section 5.1. The precise expressions are given in equations (5.24)–(5.27). The effective action of the flavored index has been calculated in many examples in the literature. In particular, the papers [19,20] discuss various examples using the Bethe-ansatz method. In each case the general expression we present in this paper agrees with the corresponding expression in the literature.
4. General saddle-point configurations
The (m, n) saddles above describe a family of saddles that can be thought of as a “string” of N eigenvalues winding around the torus. One can ask whether there are other possible shapes that the eigenvalues can take. We find a rich class of so-lutions to the saddle-point equations which can be described as follows. Consider all possible finite abelian groups of order N , the simplest such group is Z/N Z but there can be more general groups depending on the prime factors of N , see equa-tion (6.3). We find that every group homomorphism of a finite abelian group into the torus C/(Zτ + Z) (considered as an abelian group) leads to a solution of the saddle-point equations. The class of solutions that we find includes string-like solu-tions carrying Z/N Z structure that have been discussed in the literature using the Bethe-ansatz method [12,22,30,31]. The details are presented in section 6.
The plan of the rest of the paper is as follows. In section 2 we discuss the details of the elliptic extension approach and find the (m, n) string-like solutions of the large N saddle-point equations for a very general class of N = 1 superconformal quiver theories. In section 3 we calculate the action of these saddles and discuss universal solutions and the corresponding phase structure of the SCFT. In section 4 we introduce flavor chemical potentials and discuss the universal family of saddles with this refinement. In section 5
we show that, in specific domains in the space of chemical potentials, the large-N action of the (m, n) saddles takes a universal form controlled by anomalies. In section 6 we find and discuss a large family of saddle-points which are classified by finite abelian groups. In section 7 we ouline some directions of future work. In the appendices we present various technical details that are used at multiple points in the paper.
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2 Large-N saddles of quiver theories
In this section we present the superconformal index for a very general class of four-dimensional quiver gauge theories, containing matter fields in bi-fundamental or adjoint representations. In the first subsection we rewrite the index specialized to the case σ = τ in terms of a doubly-periodic non-holomorphic function associated to the torus C/(Zτ +Z). In subsections2.2,2.3we solve the saddle-point equations for the model in the continuum and the discrete formalisms, respectively, and show that a string of eigenvalues winding (m, n) times (with gcd(m, n) = 1) around the two cycles of the torus solves the saddle-point equa-tions. In subsection 2.4, we show that the contour of the original matrix integral can be deformed so as to pass through the (m, n) saddles, so that they contribute to the action.
We consider a N = 1 gauge theory defined by a quiver diagram with ν nodes labelled by the index a = 1, . . . , ν. Each node a is associated with the gauge group SU(Na), so that
the gauge group of the theory is G =Qν
a=1SU(Na). The matter multiplets are described
by arrows connecting pairs of nodes (a, b). Each arrow represents a chiral superfield, transforming in the bi-fundamental representation (Na, Nb) of SU(Na) × SU(Nb), and
having R-charge rab. This notation includes chiral superfields transforming in the adjoint representation of a gauge group factor SU(Na), with R-charges raa.
For a quiver gauge theory of this type, the index (1.2) can be represented as an integral over ν unitary matrices [6,7,32] which are interpreted as the holonomies of the gauge field factors around the S1. After integrating over angular variables, this matrix integral reduces to an integral over the eigenvalues of the matrices. Writing the eigenvalues of the unitary matrices as e2πiuai in terms of the gauge variables ua
i ∈ R/Z, the matrix integral can be
written as an integral over these gauge variables, one for each i-th direction in the a-th Cartan torus, with a certain measure factor. We use the notation u to denote the set of all gauge variables uai, i = 1, . . . , Na, a = 1, . . . , ν. The precise form of the superconformal
index (1.7) is as follows (with q = e2πiτ, Im τ > 0), I(τ ; n0) = (q; q)2Pνa=1Na Z [Du] ν Y a=1 Na Y i,j=1 i6=j Γe uaij + 2τ ; τ, τ × Y a→b Na Y i=1 Nb Y j=1 Γe uabij +rab 2 (2τ − n0); τ, τ , (2.1) with uabij = uai − ub
j, uaij = uai − uaj. Here, the first line includes the vector multiplet
contribution while the second line is the chiral multiplet contribution. The symbol Qν
a=1
denotes a product over the different gauge factors SU(Na), whileQa→bdenotes the product
over all chiral superfield contributions (namely, the contributions associated with arrows in the quiver diagram that start from any node a and reach any node b). Again, allowing the head and tail of the arrows to be identified, this notation incorporates the contributions of chiral superfields transforming in the adjoint representation of a gauge group factor SU(Na), with uaaij = uai − ua
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function Γe are defined in (A.1), (A.3), respectively. The measure of integration is [Du] = ν Y a=1 Na Y i=1 duai 1 Na! δ N X i=1 uai ! , (2.2)
and the contour of integration for each of the uai’s is R/Z for which we can choose the repre-sentative [0, 1). Note that effectively the vector multiplet associated with each gauge factor contributes as an adjoint chiral multiplet with R-charge 2. For the R-charge of the chiral multiplets we assume 0 < rab < 2,2 which is indeed satisfied for all the quiver theories that we consider, in particular for the superconformal quivers with a known supergravity dual. 2.1 The elliptic form of the action for N = 1 quiver theories
As mentioned in the introduction, in order to analyze the integral (2.1), we deform the in-tegrand so as to make it well-defined on the torus C/(Zτ + Z). The new inin-tegrand, which is doubly periodic in each eigenvalue ui, is no longer meromorphic in ui. Instead, the real and
imaginary parts are real-analytic (except for at finite number of points in the fundamental domain).3 In the large-N approximation, one has to find configurations of eigenvalues on the torus which solve the variational problem. Due to the lack of meromorphy of the inte-grand, one has to study the variational problem in both the uai and uai variables separately as the vanishing of one of these equations no longer guarantees the vanishing of the other. In order to implement the deformation, we introduce two doubly-periodic functions. The first one P (z; τ ), defined in (A.11), is closely related to the Jacobi theta function which should be reasonably familiar to most string theorists. This function has a long history starting from the 19th century (see [35]), and its Fourier expansion along its two periods is well-known as the second Kronecker limit formula (A.12). The second function Q(z; τ ) [36,
37] is relatively unknown in the physics literature, it is related to the so-called Bloch-Wigner elliptic dilogarithm [33]. This function has been studied intensively by number theorists in the last few decades and, in particular, one knows the double Fourier expansion [34] which we present in (A.18).
Using these building blocks, we construct the function
Qc,d(z) = Qc,d(z; τ ) = q c3 6 − c 12 Q(z + cτ + d) P (z + cτ + d)c , c, d ∈ R . (2.3)
This function is clearly elliptic as all its building blocks are, and it obeys the property Qc,d(z) = Γe(z + (c + 1)τ + d; τ, τ )−1 when z2= 0 . (2.4) In order to deform the integral expression for the index, one simply replaces each function Γe(z + (c + 1)τ + d; τ, τ )−1 in the integrand of (2.1) by Qc,d(z).
2This assumption ensures that there are no zeros or poles of the integrand when uab ij = 0.
3We shall call such functions doubly periodic or sometimes elliptic. This is an abuse of terminology as
usually the notation elliptic is used for meromorphic functions. Our terminology follows that of the elliptic dilogarithm [33,34] — a non-meromorphic function — which is one of the main players in the analysis.
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Following this procedure, we obtain the following expression for the integral (2.1), I(τ ) =
Z
[Du] exp −S(u)
, (2.5)
where the deformed integrand, called the elliptic action S(u), is defined as
S(u) = −2 ν X a=1 Nalog q− 1 24η(τ )+ ν X a=1 Na X i,j=1 i6=j V (uaij) +X a→b Na X i=1 Nb X j=1 Vab(uabij) , (2.6) where P
a→b denotes the sum over all chiral superfield contributions. The “potential”
functions describing the interaction between the gauge variables are given by V (z) = log Q1,0(z) = πiτ 6 + log Q(z + τ ) − log P (z + τ ) , Vab(z) = log Qcab,dab(z) = πiτ 6 2c3ab− cab + log Q(z + cabτ + dab) − cablog P (z + cabτ + dab) . (2.7)
Here the doubly-periodic functions Q(c,d) are defined as above, and cab = rab− 1 , dab = −n0
rab
2 . (2.8)
We note that the functions P and Q are invariant under the shift d 7→ d + 1, so that Qc,d+1(z) = Qc,d(z). Thus, Pi,jV (uaij) describes the contribution of the gauge superfield
at node a to the action S(u), whileP
i
P
jVab(uabij) is the contribution of a chiral superfield
associated to an arrow going from node a to node b. For definiteness, we set Vab = 0 if
there is no arrow going from a to b in the quiver diagram.
2.2 The saddle-point equations and (m, n) solutions in the continuum limit In this subsection we find stationary points for the class of N = 1 superconformal quiver theories that we considered above. We begin with the action (2.6) rewritten slightly:
S(u) = S0+ ν X a=1 Na X i,j=1 i6=j V uai − uaj +X a→b Na X i=1 Nb X j=1 Vab uai − ubj − ν X a=1 Na λa Na X i=1 uai +λea Na X i=1 ua i ! . (2.9)
Here the functions V (z) and Vab(z) are doubly periodic complex-valued functions as dis-cussed above, and S0is independent of u, ¯u.4 The function V encodes the contribution from the vector multiplets, while Vab describes the contribution of the chiral multiplets going
from node a to node b, and having R-charge rab. Since the action is not meromorphic,5 we
4We use the notation z = z∗
for the complex conjugate.
5This is sometimes denoted by having the complex conjugate of the argument as an additional variable
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have to solve the saddle point equations for uai and ua
i separately. The Lagrange multipliers λa,λea, implement the SU(Na) constraints on the full complexified gauge holonomies, i.e.,
Na X i=1 uai = 0 , Na X i=1 ua i = 0 . (2.10)
Note that λa,λea are a priori independent variables as we are allowing for complex saddles and the extended action is not meromorphic. Here we have defined the Lagrange multipliers with a factor of Na in anticipation of the fact, that we will see below, that the value of λa
is O(1). In principle we could define a large-N limit by keeping the different values of Na distinct and taking all of them large in some specified way. However for simplicity we will assume Na= N for all a, and then take N large. We note that there is no obstruction to carrying this analysis in the general case. The superconformal quivers that we are mainly interested in do satisfy this condition.
In the large-N limit, it is convenient to pass to the continuum formulation by using the following identifications at each node
i N 7→ x , 1 N 7→ dx , u a i 7→ ua(x) , N X i=1 7→ N Z 1 0 dx , (2.11)
where x ∈ [0, 1). In this way the action (2.9) becomes the functional 1 N2 S[u] = ν X a=1 Z 1 0 dx Z 1 0 dy V ua(x) − ua(y) +X a→b Z 1 0 dx Z 1 0 dy Vab ua(x) − ub(y) − ν X a=1 λa Z 1 0 dx ua(x) +λea Z 1 0 dx ua(x) . (2.12)
Notice that we have dropped the term S0, as it is subleading at large N . On the other hand, we cannot drop the Lagrange multiplier term.
Let us discuss the extremization equations. Varying with respect to ua(x) gives Z 1
0
dy [∂V (ua(x)−ua(y))−∂V (ua(y)−ua(x))]
+ X fixed a→b Z 1 0 dy ∂Vab(ua(x)−ub(y))− X fixed a←b Z 1 0 dy ∂Vba(ub(y)−ua(x))−λa= 0 , (2.13)
where ∂ denotes the holomorphic derivative with respect to the argument of the func-tion, and the sums are over all chiral fields that go from the fixed node a to any node b (“fixed a → b”), or that reach the same node a starting from any node b (“fixed a ← b”). The equations arising from varying ¯ua(x) have the same form as (2.13) with the replace-ment ∂V → ¯∂V and similarly with ∂Vab. Note that ¯∂V (u) 6= ∂V (u), and so these equations
are genuinely independent equations. Moving on, varying with respect to the Lagrange multipliers λa,eλa yields the constraints
Z 1 0 dx ua(x) = 0 , Z 1 0 dx ua(x) = 0 , (2.14)
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meaning that the unimodularity constraint is imposed on both the real and the imaginary part of the gauge variables ua(x).
We now show that the gauge variable configuration ua(x) = x T − T
2 (2.15)
is a saddle of the large-N action for any period T of the action (2.9). These periods corre-spond to the points of the lattice which are labelled by two integers (m, n). Equivalently, the uniform distribution (2.15) from 0 to mτ + n can be thought of as the uniform dis-tribution wrapping (m, n) times around the two cycles of the torus C/(Zτ + Z). In order to count independent configurations in the large-N limit, we should consider lattice points with the addition condition gcd(m, n) = 1.
Now, obviously (2.15) solves the constraint (2.14). We now show that it also satisfies the equation (2.13). Plugging (2.15) in (2.13), we obtain
Z 1 0 dy " ∂V (T (x−y))−∂V (T (y −x))+ X fixed a→b ∂Vab(T (x−y))− X fixed a←b ∂Vba(T (y −x)) # = λa. (2.16) This equation is of the form
Z 1
0
dy fa(y − x) = λa, (2.17)
where the integrand fa is periodic under the shift of the real variable y → y + 1. Since we
are integrating over the full period, the result of the integral is simply Z 1
0
dy fa(y) = λa, (2.18)
which does not depend on x. Thus we obtain the value of the Lagrange multiplier λa to be λa= Z 1 0 dy " ∂V (−T y) − ∂V (T y) + X fixed a→b ∂Vab(−T y) − X fixed a←b ∂Vba(T y) # = Z 1 0 dy " X fixed a→b ∂Vab(T y) − X fixed a←b ∂Vba(T y) # . (2.19)
Here, to reach the second line we use the fact that the integral of ∂V (−T y) equals the integral of ∂V (T y) because of periodicity. (For a periodic function f with period 1, we haveR1
0 f (y)dy = R1
0 f (y−1)dy, which is equal to R1
0 f (−y
0)dy0by the change of variable y0 = 1 − y.) Similarly the integral of ∂Vab(−T y) equals the integral of ∂Vab(T y) for the same reason. Note that λa = O(1), as anticipated. The equations arising from varying with respect to ¯ua and λea are solved in exactly the same way, withλ being determined ase
e λa = Z 1 0 dy " X fixed a→b ¯ ∂Vab(T y) − X fixed a←b ¯ ∂Vba(T y) # , (2.20)
which in general is not the complex conjugate of (2.19).
In some special cases, one may find that the expressions (2.19), (2.20) vanish, hence λa=eλa= 0 at the extremum; this means that the extremization equations are also solved for quivers with U(N ) gauge groups, and not just SU(N ).6 For instance, this happens
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for non-chiral quivers, where for every arrow going from node a to node b leading to the potential Vab, there is an arrow going from node b to node a, with identical potential Vba = Vab; this implies that two terms in the last line of (2.19) cancel against each other.
One finds λa =λea= 0 also for chiral quivers where the R-charges of bifundamental chiral multiplets are all the same (ra,b = r for all a 6= b). In this case, Vab = Vr for all a 6= b,
hence the expression for the Lagrange multiplier becomes λa= (nout,a− nin,a)
Z 1
0
dy ∂Vr(T y) , (2.21)
where nout,a is the number of arrows going out of node a, while nin,a is the number of arrows pointing towards node a. Now, cancellation of the gauge anomaly implies that at each node of the quiver the number of outgoing arrows equals the number of ingoing arrows, that is nout,a− nin,a= 0, a = 1, . . . , ν, thus showing that the saddle-point equations are solved with λa = 0. The same argument leads to λea = 0. Examples of chiral quivers where the R-charges are all equal are provided by the Yp,p and Yp,0 infinite families [38], the former being Z2p orbifolds of N = 4 SYM, and the latter being Zp orbifolds of the
conifold theory [39]. 2.3 The discrete case
We can also offer a finite-N , discrete version of the continuum discussion given above, which is useful later. The main steps are the same, so we will be more brief. We show that the gauge variable configuration
uai = T i N − N + 1 2N , i = 1, . . . , N , a = 1, . . . , ν , (2.22) extremizes the finite-N action (2.9). Clearly (2.22) obeys the constraint
N
X
i=1
uai = 0 , (2.23)
arising from the variation of (2.9) with respect to the Lagrange multiplier λa, as consistent with the SU(N ) gauge group. Varying with respect to uai, we obtain the following saddle-point equations N X j=1 ∂Va(uai − uaj) − ∂Va(uaj − uai) + X fixed a→b ∂Vab uai − ubj − X fixed a←b ∂Vba ubj− uai ! = λa, (2.24)
and then plugging (2.22) in, yields
N X j=1 ∂Va T N(i − j) − ∂Va T N(j − i) + X fixed a→b ∂Vab T N(i − j) − X fixed a←b ∂Vba T N(j − i) ! = λa. (2.25)
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We can now exploit the fact that the functions are periodic under i → i + N (because this sends uai → ua
i+ T , and all terms are T -periodic when seen as functions of u) together with
the fact that we are summing over all j = 1, . . . , N , to argue that the left hand side does not depend on the value of i, and that we can change −j into +j in the first and third term. We thus arrive at
N X j=1 X fixed a→b ∂Vab T Nj − X fixed a←b ∂Vba T Nj ! = λa, (2.26)
which just fixes the value of the Lagrange multiplier λa. Again, the equations for ¯λaand ¯uai are solved in an analogous manner.
Although we do not take a large-N limit in solving the saddle-point equations in the discrete method, we note that the validity of the saddle-point approximation to the original matrix integral needs a large-N limit; this gives the same result as the continuum limit described above. Instead of using Lagrange multipliers, we can equivalently satisfy the SU(Na) constraint by explicitly solving (2.10) for, say, uaNa in terms of the other
eigenvalues from the very beginning, and then extremizing with respect to the remaining variables. This leads us to the same final result as the procedure above.
2.4 The contour deformation
We have shown above that the uniform distribution of the gauge variables between 0 and the lattice point mτ +n, m, n ∈ Z solves the saddle-point equations of the matrix integral (2.5). In order to show that these configurations contribute to the integral, we also need to show that the contour of integration passes through the saddle-point. A contour deformation argument is not a priori obvious because the integrand of (2.5) is not meromorphic. The discussion below is an adaption of the procedure used in [21] for N = 4 SYM to the class of N = 1 theories that we discuss in this paper.
The main point is to use the interplay between the two representations of the supercon-formal index: (2.1) whose integrand is meromorphic, and (2.5) whose integrand is doubly periodic. Both these integrals are defined using the same contour in which the variables uai go from 0 to 1 on the real axis. Since the integrand of (2.1) is meromorphic, we can deform its contour without changing the value of the integral as long as we do not cross any poles of the integrand.7 Following this idea, we deform the contour of the meromorphic inte-grand to a new contour which passes through a given saddle, and then show that on this new contour we can replace the meromorphic integrand by the doubly-periodic integrand without changing the value of the integral at large N .
As explained in [21], the new contour C consists of three pieces in each variable uia, which we denote as Chor + Cvert+ Csaddle. The piece Chor runs over a subset of the real axis, here equation (2.4) shows that the two integrands agree. The piece Cvert consists of two closely placed oppositely oriented vertical lines, and the integral along this piece of either of the two integrands vanishes (and therefore the replacement is valid). The
7The residues picked up from crossing of these poles could lead to important physical phenomena. We
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third piece Csaddle, which is the non-trivial piece, is an infinitesimal horizontal strip passing through the saddle-point value of uai. It was shown in [21] that the value of the two integrands for N = 4 SYM agree at the saddle-point value, and that consequently one can make the replacement in an infinitesimal small neighborhood of the saddle-point to good approximation. One then uses the saddle-point approximation on the new contour so that the value of the integral is the value of the integrand at the saddle-point in the leading large-N approximation.
The part of the argument that depends in a non-trivial manner on the theory under consideration is the agreement of the meromorphic and the doubly-periodic action when evaluated on the saddle-point. As we now show, this holds generically for the N = 1 theories discussed here. We recall, from the discussion in the previous subsections (in particular, see equation (2.22)), that the (m, n) saddle point of the action (2.5) is described by the following gauge variable configuration
uai = (mτ + n) i
N + u0 ≡ ui, i = 1, . . . , N , a = 1, . . . , ν , (2.27) with the value of the constant u0 chosen so as to obey the SU(N ) constraint.
We start by recalling a relation, that involves the doubly periodic functions P and Q, and the elliptic Gamma function Γe [21,36,37],
Q(z; τ ) = e2πiαQ(z1,z2)q13B3(z2)−12z2B2(z2) P (z; τ ) z2
Γe(z + τ ; τ, τ ), (2.28) where the function αQ is a real function of z1 and z2 which is not doubly periodic. The function αQ can be written as a sum of an explicit non-periodic function and a doubly-periodic function8 ΨeQ (to be determined below), as follows [21],
αQ= −
1
4(1 + 2 {z1})bz2c (1 + bz2c) + 1
2ΨQ(z1, z2) . (2.29) The function αQ−12ΨQ is piecewise continuous and it vanishes in the region −1 ≤ z2 < 1. Upon substitution of the function Q as given in equation (2.28), in the definition of the function Qc,d in terms of Q and P , as given in equation (2.3), it follows that
Qc,d(z) = e2πiαQ(z1+d, z2+c)q−Ac(z2)
P (z + (c + 1)τ + d; τ )z2
Γe(z + (c + 1)τ + d; τ, τ ), (2.30) the cubic polynomial Ac is
Ac(x) = 1 6x 3+1 2c x 2+1 2c 2x − 1 12x . (2.31)
The doubly-periodic action (2.6), (2.7) is a linear combination of the functions Qc,d,
evaluated on the gauge variables. Each one of the summands in (2.6) corresponds to a
8We recall that the Fourier expansions of the doubly periodic functions P and Q defined in
equa-tions (A.17) and (A.18), have implicit ambiguities that we parameterize by two real and doubly periodic functionsΨeP andΨeQ, respectively. The function ΨQis determined by these two functions. To be concrete,
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specific multiplet. We show below that after summing over all the gauge variables in the ansatz (2.27) and over all the matter multiplets, the contributions coming from the polynomial Ac(z2), and the function z2log P (z + (c + 1)τ + d; τ ) vanish. Thus we reach the conclusion that the absolute value of the integrands of (2.1) and (2.5) are equal on the (m, n) saddle point configurations. Next we choose the phase ΨQ such that the phases
of the doubly periodic and meromorphic integrands are also equal when evaluated on the (m, n) saddles.9
First we analyze the contribution that comes from the cubic polynomial Ac given
in (2.31). The integrand in question involves a product over all supermultiplets in the theory, that here we label by an index α (this includes the vector multiplet). Each factor contributes with a corresponding polynomial Acα. Let ρ
(a)
α denote the weights of the
representation R(a)α that the supermultiplet α carries under the gauge group at the a-th node
of the quiver. After summing over all the weights ρ(a)α and then over all the supermultiplets,
the contributions coming from the four terms on the right-hand side of (2.31) can be organized in linear combinations of the following four expressions,
X α X ρ ρ(a)iα ρ(b)jα ρ(c)kα ! ua2iub2juc2k, X α X ρ (rα− 1) ρ(a)iα ρ(b)jα ! ua2iub2j, X α X ρ (rα− 1)2ρ(a)iα ! ua2i, X α X ρ ρ(a)iα ! ua2i. (2.32)
Here the indices i, j and k are summed over all possible values, while the indices a, b and c la-beling the nodes of the quiver are kept fixed. Finally, (rα−1) is the R-charge of the fermion
field in the multiplet α (we formally assign rα= 2 to the vector multiplet, so that the gaug-ino has the correct R-charge 1). The sum over ρ means that one needs to sum over all the weights ρ(a)α that belong to the representation R(a)α . The u-independent terms in (2.32)
are the Gauge-Gauge-Gauge, R-Gauge-Gauge, R-R-Gauge and mixed Gauge-gravitational anomaly coefficients, respectively, for the Cartan generators of the gauge group. These
9
Here a question arises as to whether this prescription for ΨQ is well-defined. In particular, it could
happen that a certain point z on the torus lies on the string of eigenvalues for two different saddles (m, n) and (m0, n0). The point z would correspondingly lift to two different points in the complex plane which differ by a lattice translation. The question then is whether the value of the phase of Γeand in particular
the value of αQagrees at these two points. This is a subtle question whose complete analysis will be posted
elsewhere. For our purposes here, we restrict our analysis to a set of saddles with an upper cutoff on m. In this situation if we take the first term in the right-hand side of (2.29), the difference of evaluating this between two points differing by a lattice translation, is a rational number with a bounded denominator. We can then lift our discussion to a larger torus (which is still finite) on which ΨQ is well-defined. We
note that all the calculations of the action are done by considering configurations of gauge variables that are extended on the complex plane (not just restricted to the fundamental domain), so that they are not affected by this cutoff.
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vanish in anomaly-free theories that have an R-symmetry conserved at the quantum level, as we assume here. Thus we conclude that the contribution of the cubic polynomial Ac to
the integrand vanishes.
Then we move to the function z2log P . The contribution to the action of the function P associated to a given multiplet can be written as the exponential of
N
X
i,j=1
(uij)2log P uij + (c + 1)τ + b
. (2.33)
We can evaluate this expression on the saddle point ui = Ni (mτ + n) + u0 using the double Fourier expansion (A.17) for the function log P . In this manner we obtain a sum over the integers en,m of terms that are proportional toe
N X i,j=1 (i − j) e i − j N (nm −e mn)e , (2.34)
where we are using the notation e(x) = e2πix. These terms can be proven to vanish as follows. Let us define k =enm −mn thene
N X i,j=1 (i − j) e i − j N k = N X i=1 i e i Nk N X j=1 e −j N k − N X j=1 j e −j N k N X i=1 e i Nk = N X i=1 i e i Nk δk,0− N X j=1 j e −j N k δk,0 = δk,0 N X i=1 i − N X j=1 j = 0 . (2.35)
Let us recapitulate the procedure that we followed. We begin with the meromorphic integral (2.1) whose contour can be deformed freely up to potential residues. Then we argue that there exists a contour which passes through the (m, n) configuration such that the value of the meromorphic integral (2.1) equals the value of the doubly-periodic integral (2.5) along the contour. Since we have already checked that the (m, n) configurations solve the saddle-point equations of the doubly-periodic action separately for the real and imaginary parts, we use the doubly-periodic action to implement the saddle-point approximation. This leads to the conclusion that the integral on that contour is dominated by the value of the integrand in the vicinity of the saddle. We stress that a rigorous global analysis remains to be done.10 Such an analysis is outside the scope of this paper. In section 3.4
we perform a naive analysis of relative dominance of the saddles.
3 The effective action of the (m, n) saddle
In this section we compute the action of the large-N saddles (2.15) with period T = mτ +n. The action that has the least real part will dominate and thus provide our estimate for the index (2.5) in the grand-canonical ensemble, wherein the angular chemical potential τ is the independent variable.
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3.1 Evaluation of the actionUpon evaluating the continuum action (2.12) on the configurations (2.15), one obtains the large-N effective action
Seff(m, n; τ ) = νN2 Z 1 0 dx Z 1 0 dy V T (x − y) + N2X a→b Z 1 0 dx Z 1 0 dy Vab T (x − y), (3.1)
which depends on the complex parameter τ as well as on the integers m, n that appear in T = mτ + n. We can reduce each double integral to a single integral as follows,
Z 1 0 dy Z 1 0 dx V T (x − y) = Z 1 0 dy Z 1 0 dx V T x = Z 1 0 dx T x , (3.2)
where we have used periodicity of the potential in establishing the first equality. Recalling the definitions (2.7), we obtain
Seff(m, n; τ ) = νN2 Z 1 0 dx log Q1,0((mτ +n)x)+N2 X a→b Z 1 0 dx log Qcab,dab((mτ +n)x) , (3.3)
where each function Qc,d denotes the contribution of a chiral multiplet, and Q1,0 is the contribution of the SU(N ) vector multiplet. Evaluating these integrals using formulae provided in appendixAwe reach our final expression for the large-N action, to be presented below. One can see that the result does not depend on any common divisor of m and n. Also, notice from (3.1) that Seff(−m, −n; τ ) = Seff(m, n; τ ), since a change of sign T → −T just amounts to swapping the integration variables. Hence without loss of generality from now on we assume that m and n are relatively prime, with m ≥ 0.
The saddle m = 0, n 6= 0. We first discuss the special case m = 0, n 6= 0, where the gauge variables take real values ua(x) = nx −12. In the large-N limit, the eigenvalue distribution of all these saddles on the torus C/(Zτ + Z) are equivalent. Recalling that we assume that the R-charges of all chiral multiplets satisfy 0 < rab < 2, the identity (A.23)
implies that the real part of the action vanishes at order O(N2). This saddle in the form (m, n) = (0, 1) corresponds to the saddle discussed in [7]. Indeed, in the saddle of [7] the gauge variables — which are assumed to be real — take the uniform density ρ(u) ≡ dudx = 1 which corresponds to u(x) = x + constant, and the corresponding action is independent of N at leading order.
From now on we take m > 0. Evaluating the doubly-periodic potentials (2.7) at the saddles and using Identities (A.14), (A.19), we find that the effective action (3.1) can be expressed in terms of Bernoulli polynomials
B2(z) = z (z − 1) + 1 6, B3(z) = z z −1 2 (z − 1) , (3.4)
depending on the variable
zab = {mdab− ncab} = −(n0m + 2n) rab 2 , (3.5)
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where in the second equality we used (2.8), and for any real x we define the fractional part {x} = x − bxc.11 The action then reads
Seff(m, n; τ ) = πiτ 6 2TrR3−TrR+ πi m(mτ +n) " TrR 6 +N 2X a→b (rab−1) B2(zab)− 1 6 # + πiN 2 3m(mτ +n)2 X a→b B3(zab)+πiN2Φ , (3.6)
where Φ is a real τ -independent function that we discuss below. Before that we note that the expression (3.6) can be rewritten in a compact form by using the following identity involving Bernoulli polynomials,
B3(x + y) = B3(x) + 3B2(x)y + 3B1(x)y2+ y3, x, y ∈ C . (3.7) Applying this to the right-hand side of (3.6) we obtain
Seff(m, n; τ ) = πiN2 3m(mτ + n)2 " ν B3(mτ + n) + X a→b B3 zab+ (mτ + n)(rab− 1) # − πiτ 6 TrR + πiN 2(Ω + Φ) , (3.8) where N2Ω ≡ − n 3mTrR 3+N2 2m " ν +X a→b (rab− 1)2 1 − 2 −(n0m + 2n) rab 2 # (3.9)
is τ -independent and purely real. This rewriting will be useful in section 4.
In eq. (3.6), the term linear in τ is the result of resumming the corresponding terms in (2.7) into the R-symmetry anomaly coefficients (B.1)
νN2+ N2 X a→b 2(rab− 1)3− (rab− 1) = 2TrR3− TrR = 16 9 (a + 3 c) . (3.10) The last equality in (3.10) shows the combination of a and c Weyl anomaly coefficients that is obtained using the relations (B.2) for superconformal theories. We remark that this term is proportional to the supersymmetric Casimir energy on a round S3× S1 [43,
44]. For the SU(N ) quivers we are considering, cancellation of the R-Gauge-Gauge ABJ anomaly implies TrR = 0 at leading O(N2) order, see appendix B for details. However we temporarily keep the TrR term in the result with the purpose of showing a remarkable agreement with the Cardy-like limit of the index at finite N , to be discussed momentarily.
11We note that B
2({x})−16 = −{x}(1−{x}) ≡ −ϑ(x) and B3({x}) =12{x} (1 − {x}) (1 − 2{x}) ≡ 12κ(x),
where ϑ(x) and κ(x) are the functions used e.g. in [13, 14, 17, 42]. Some more details on Bernoulli polynomials are given in appendixA.
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3.2 The τ -independent part of the actionThe term Φ appearing in (3.6) is a real, τ -independent function of m, n, as well as of the number of nodes ν and the R-charges rab, that remains not determined by our technology
as it has been developed so far. It arises from the Fourier modes of the function ΨQ(z) discussed in section 2.4, see appendixA for its definition from the integrals in (3.3). Fol-lowing the discussion in section 2.4, the constant Φ should be determined by demanding that Seff(m, n; τ ) matches the meromorphic extension of the integrand of (2.1), evaluated on the gauge variable configurations (2.15). In the rewriting (3.8) the terms Φ and Ω are naturally combined, as they are both real and τ -independent, and in section 4 we will see that a comparison with other results in the literature indeed relates Φ to Ω.
Before continuing, we discuss to what extent the value of Φ affects the results of our analysis, in particular in relation to the comparison with the gravity side. Since it yields a purely imaginary, τ -independent contribution to the action, Φ is not relevant for the phase structure of the index in the grand-canonical ensemble, in the case where only one saddle dominates. Indeed in the grand-canonical ensemble, for each value of the chemical potential τ the dominating large-N saddle is the one with least real part of the action Seff, and the corresponding value of the partition function is log Zgrand= −Seff.12 If instead we discuss the microcanonical ensemble, where the large-N partition function is given by the Legendre transform log Zmicro = (τ ∂τ− 1)Seff, things are more subtle. Being independent of τ , πiN2Φ appears in the Legendre transform (1 − τ ∂τ)Seff precisely in the same way as it appears in Seff, and thus just contributes to the imaginary part of log Zmicro. While a priori one could imagine discarding the imaginary part and regarding the entropy as the real part of log Zmicro, it has been shown [8–10] that the correct procedure is more delicate. In fact one should impose the vanishing of the imaginary part of log Zmicro in order to reproduce the O(N2) entropy of known supersymmetric AdS5 black hole solutions. This means that Φ would play a relevant role, as it appears in Im(log Zmicro) = 0. As illustrated in [9], the latter condition corresponds to a constraint on the J and Q variables in the supersymmetric microcanonical ensemble. Relatedly, after imposing the constraint the expectation values for J and Q in the grand-canonical ensemble depend on Φ. It appears that only a specific choice of Φ gives the correct charges that match the dual gravitational solution. One way to fix Φ that is in agreement with the gravitational results is to regard the action Seff as a holomorphic function of the chemical potentials τ and ϕ that are conjugate to the angular momentum 2J+and to the R-charge Q, respectively, before imposing the relation ϕ = τ −n20 that leads to the index (1.7).13 We will discuss a concrete example in the next subsection.
The upshot is that despite the fact that in our treatment we have not determined the form of Φ, we have argued that in principle it should be determined by demanding that
12
If there are multiple saddles that have the same minimum value of Re(Seff), then knowing the phase of
their exponential contributions e−Seff to the index becomes crucial to determine how they are resummed.
In this case Φ plays an important role. However in order to determine this phase we would need to know the subleading corrections to the large-N limit, which is out of the scopes of the present work. See [12] for a discussion of this phenomenon in the present context.
13This would lead us slightly off the supersymmetric sector that is captured by the index. Related to this
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Seff(m, n; τ ) matches the meromorphic extension of the integrand of (2.1). On the otherhand, for saddles that can be compared with gravity solutions, there is a distinguished value of Φ that leads to a complete matching of the results on the two sides. In the following sections we will see that these two choices are not entirely in agreement.
3.3 Special families of saddles
We further analyze the structure of the saddle-point action (3.6) by identifying some notable cases. In general this depends on the details of the quiver considered. A convenient way to classify the (m, n) saddles in view of evaluating the action is to consider the families defined by the different values of the integer
` ≡ −n0m − 2n (3.11)
which appears in the argument
zab= n `rab 2 o (3.12) of the Bernoulli polynomials in (3.6). Here we always assume m > 0, and we recall that m, n are coprime. The evaluation of the action is straightforward if
− 1 < `rab
2 < 1 for all rab, (3.13)
as in this case we can trivially take care of the fractional part for all chiral superfields and evaluate the Bernoulli polynomials in (3.6). This condition is obviously satisfied for ` = 0 and, since we assume that all R-charges lie in the range 0 < rab < 2, by ` = ±1. We will see that the families of saddles characterized by ` = 0, ±1 lead to an action which is universal, in the sense that it depends on the field theory data only through R-symmetry anomaly coefficients.
If all chiral multiplets have R-charge r = 2/3, as for N = 4 SYM and its orbifolds, then z = {`3} is determined by ` (mod 3) and can only take the values z = 0,1
3, 2
3. In this case evaluation of the action is straightforward for any choice of `, see [21] for a thorough analysis of this case. For more general theories this is not true. If the R-charge r of a given chiral multiplet is rational, then there will be finitely many possible values of the corresponding variable z,14while for the generic case where the R-charge is irrational there are no equivalent choices of ` and one has infinitely many possible values of z, which makes a detailed study of the action complicated.
If all R-charges satisfy 0 < rab< 1, which is true for many theories, it is also straightfor-ward to evaluate the action of saddles such that ` = 2, however as we will see its expression is not entirely captured by anomaly coefficients. For |`| > 2, the condition (3.13) is not satisfied in a generic theory, so the analysis becomes case-dependent and we will not discuss it further.
After illustrating these consideration in more detail, below we study the case of the conifold theory as a simple example where the exact superconformal R-charges are rational and the complete phase structure of the (m, n) saddles can be worked out.
14Say r = p/q, with p, q relatively prime. Then z is determined by ` (mod q) if p is even, and by ` (mod 2q)
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The family ` = 0. When n0 is odd, the coprime integers solving the condition ` = 0are (m = 2, n = −n0), while if n0 is even we need to take (m = 1, n = −n0/2). In both cases, the action (3.6) evaluates to
Seff(` = 0; τ ) = πiτ 6 2TrR3− TrR+ πi TrR 6m(mτ + n) + πiN 2Φ , (3.14)
where we used the definition of TrR in (B.1) at large-N . Since this saddle-point action is guaranteed to be correct only at O(N2) order, we should set TrR = 0 as a consequence of the R-Gauge-Gauge anomaly cancellation for the quivers of interest in this paper. This leaves us with
Seff(` = 0; τ ) = πiτ
3 TrR
3+ πiN2Φ . (3.15)
Notice that the Legendre transform of this Seff is purely imaginary, (1 − τ ∂τ)Seff = πiN2Φ. We conclude that, independently of the value of Φ, these saddles carry no O(N2) entropy. We can also compare the expression above with the Cardy-like limit of the index. Take n0 = 0, namely consider the standard index with no shift of the angular chemical potential, cf. (1.2). Then we have (m = 1, n = 0), and upon taking the small-τ limit the leading order TrR/τ term in (3.14) remarkably agrees with the Cardy-like formula of [48], which is derived at finite-N . This indicates that (3.14) correctly captures at least a part of the finite-N action of the (m, n) saddles considered here. We leave the analysis of the subleading corrections to the O(N2) result for future work.
The family ` = ±1. In this case both n0 and m are odd. In particular, this family of saddles exists for the index (1.6) but not for the n0 = 0 index. Also recall that we assume that all R-charges satisfy 0 < rab < 2. Then the Bernoulli polynomials evaluate to
B2(zab) = 1 4rab(rab− 2) + 1 6, B3(zab) = ± 1 8rab(rab− 1)(rab− 2) , (3.16) where the sign choice is correlated with ` = ±1. Plugging this in the effective action (3.6) and using X a→b N2rab(rab− 1)(rab− 2) = TrR3− TrR , (3.17) we arrive at Seff(` = ±1; τ ) = πiτ 6 2TrR3− TrR+ πi 12m(mτ + n) 3TrR3− TrR ± πi 24m(mτ + n)2 TrR3− TrR+ πiN2Φ . (3.18)
Before omitting the TrR terms, let us consider the sub-case n0= ∓1, m = 1, n = 0, in which case the action (3.18) reads
1 πiSeff(m = 1, n = 0, n0 = ∓1; τ ) = 8τ3+ 6τ − n 0 24τ2 TrR 3−2τ − n0 24τ2 TrR − τ 6TrR + N 2Φ . (3.19)
JHEP11(2020)150
Upon taking the Cardy-like limit τ → 0, we can compare the O(τ−2) and O(τ−1) terms in this expression with the results of [16,17], finding perfect agreement. The asymptotic methods used in [17] hold at finite-N but only provide the divergent terms in the small-τ regime. Here we find complementary results which hold at leading order in the large-N expansion but are valid at finite τ . We emphasize again that for the quivers considered, the terms proportional to TrR are subleading in the large-N limit and thus in principle we have no control on them, nonetheless it is remarkable that all the O(τ−2) and O(τ−1) terms in (3.19) agree with the finite-N results obtained from the Cardy limit of the index. As for the ` = 0 saddle, it would be interesting to clarify to what extent the action (3.19) captures the finite-N contribution to the index from these saddles.
Let us now restrict ourselves to the leading O(N2) order and hence take TrR = 0 and discuss the comparison with the gravity side. We observe that we can rewrite (3.18) in the suggestive form Seff(` = ±1; τ ) = πi 24 (2mτ + 2n ± 1)3 m(mτ + n)2 TrR 3− πi 6m(2n ± 3)TrR 3+ πiN2Φ , (3.20)
and that the choice
N2Φ`=±1 = 2n ± 3 6m TrR
3 (3.21)
gives the action the form of a “perfect cube”, namely
Seff(` = ±1; τ ) = πi 24 (2mτ + 2n + `)3 m(mτ + n)2 TrR 3= πi 24 (2τ − n0)3 (τ + n/m)2 TrR 3. (3.22)
For superconformal theories we can convert the R-symmetry anomaly coefficient into Weyl anomaly coefficients setting 329 TrR3 = a = c, which yields
Seff(` = ±1; τ ) = 4πi a
27
(2τ − n0)3
(τ + n/m)2 . (3.23)
For N = 4 SYM, this agrees with the result found in [21].15
The expression (3.23) provides a prediction for the on-shell action I of putative dual supergravity solutions, which would compete in the semiclassical approximation to the gravitational path integral in the same way as the saddles compete in the large-N expression for the superconformal index. For definiteness, let us assume the theory is dual to type IIB supergravity on Sasaki-Einstein manifolds. Using the dictionary a = c = πL8G3
5, where
L is the AdS5 radius and G5 the five-dimensional Newton constant, we obtain I(` = ±1; τ ) = iπ
2L3 54G5
(2τ − n0)3
(τ + n/m)2 . (3.24)
For m = 1, n = 0, and ` = −n0 = ±1 this matches the supergravity on-shell action of supersymmetric AdS5 black holes computed using the prescription of [9]. For more general
15This is given in equation (4.13) of [21]. To compare, one should recall that for N = 4 SYM at large-N a = c = N2
4 , and that the Dirichlet character χ1(−n0m+n) appearing there evaluates to χ1(−n0m+n) = +1
